Exam 8: Confidence Interval Estimation

The t distribution is used to develop a confidence interval estimate of the population mean when the population standard deviation is unknown.

TABLE 8-3 A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine. The desired length of the insulation is 12 feet. It is known that the standard deviation in the cutting length is 0.15 feet. A sample of 70 cut sheets yields a mean length of 12.14 feet. This sample will be used to obtain a 99% confidence interval for the mean length cut by machine. -Referring to Table 8-3, the confidence interval is valid only if the lengths cut are normally distributed.

TABLE 8-3 A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine. The desired length of the insulation is 12 feet. It is known that the standard deviation in the cutting length is 0.15 feet. A sample of 70 cut sheets yields a mean length of 12.14 feet. This sample will be used to obtain a 99% confidence interval for the mean length cut by machine. -Referring to Table 8-3, the confidence interval indicates that the machine is not working properly.

TABLE 8-5 The actual voltages of power packs labeled as 12 volts are as follows: 11.77, 11.90, 11.64, 11.84, 12.13, 11.99, and 11.77. -Referring to Table 8-5, a confidence interval for this sample would be based on the t distribution with _______ degrees of freedom.

A point estimate consists of a single sample statistic that is used to estimate the true population parameter.

TABLE 8-12 The president of a university is concerned that illicit drug use on campus is higher than the 5% acceptable level. A random sample of 250 students from a population of 2000 revealed that 7 of them had used illicit drug during the last 12 months. -Referring to Table 8-12, what is the critical value for the 90% one-sided confidence interval for the proportion of students who had used illicit drug during the last 12 months?

It is desired to estimate the average total compensation of CEOs in the Service industry. Data were randomly collected from 18 CEOs and the 97% confidence interval was calculated to be ($2,181,260, $5,836,180). Which of the following interpretations is correct?

TABLE 8-4 To become an actuary, it is necessary to pass a series of 10 exams, including the most important one, an exam in probability and statistics. An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program. They take a sample of 8 actuarial students in this program and determine that their scores are: 2, 5, 8, 8, 7, 6, 5, and 7. This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program. -Referring to Table 8-4, for the confidence interval to be valid, it is necessary that test scores of students in the special study program on the actuarial exam be normally distributed.

TABLE 8-15 A sales and marketing management magazine conducted a survey on salespeople cheating on their expense reports and other unethical conduct. In the survey on 200 managers, managers have caught salespeople cheating on an expense report 58% of the time, working a second job on company time 50% of the time, listing a "strip bar" as a restaurant on an expense report 22% of the time, and giving a kickback to a customer 19% of the time. -Referring to Table 8-15, construct a 95% confidence interval estimate of the population proportion of managers who have caught salespeople giving a kickback to a customer.

A quality control engineer is interested in estimating the proportion of defective items coming off a production line. In a sample of 300 items, 27 are defective. A 90% confidence interval for the proportion of defectives from this production line would go from ____to____ .

TABLE 8-14 The president of a university is concerned that the percentage of students who have cheated on an exam is higher than the 1% acceptable level. A confidential random sample of 1000 students from a population of 7000 revealed that 6 of them said that they had cheated on an exam during the last semester. -Referring to Table 8-14, using on the 90% one-sided confidence interval, the superintendent can be 85% confident that no more than 1% of the students at the university had cheated on an exam during the last semester.

TABLE 8-4 To become an actuary, it is necessary to pass a series of 10 exams, including the most important one, an exam in probability and statistics. An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program. They take a sample of 8 actuarial students in this program and determine that their scores are: 2, 5, 8, 8, 7, 6, 5, and 7. This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program. -Referring to Table 8-4, if we use the same sample information to obtain a 95% confidence interval, the resulting interval would be narrower than the one obtained here with 90% confidence.

A major department store chain is interested in estimating the average amount its credit card customers spent on their first visit to the chain's new store in the mall. Fifteen credit card accounts were randomly sampled and analyzed with the following results: $\overline{X} = \$50.50$ and s² = 400. Construct a 95% confidence interval for the average amount its credit card customers spent on their first visit to the chain's new store in the mall assuming that the amount spent follows a normal distribution.

The difference between the lower limit of a confidence interval and the point estimate used in constructing the confidence interval is called the sampling error.

An economist is interested in studying the incomes of consumers in a particular region. The population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in an average income of $15,000. What is the upper end point in a 99% confidence interval for the average income?

TABLE 8-14 The president of a university is concerned that the percentage of students who have cheated on an exam is higher than the 1% acceptable level. A confidential random sample of 1000 students from a population of 7000 revealed that 6 of them said that they had cheated on an exam during the last semester. -Referring to Table 8-14, what is the critical value for the 90% one-sided confidence interval for the proportion of students who had cheated on an exam during the last 12 months?

TABLE 8-13 A university wanted to find out the percentage of students who felt comfortable reporting cheating by their fellow students. A surveyed of 2,800 students was conducted and the students were asked if they felt comfortable reporting cheating by their fellow students. The results were 1,344 answered "yes" and 1,456 answered "no." -Referring to Table 8-13, a confidence interval estimate of the population proportion would only be valid if the distribution of the number of students who feel comfortable reporting cheating by their fellow students is normal.

TABLE 8-1 A random sample of 100 stores from a large chain of 1,000 garden supply stores was selected to determine the average number of lawnmowers sold at an end-of-season clearance sale. The sample results indicated an average of 6 and a standard deviation of 2 lawnmowers sold. A 95% confidence interval (5.623 to 6.377) was established based on these results. -Referring to Table 8-1, we do not know for sure whether the true population mean is between 5.623 and 6.377 lawnmowers sold.

TABLE 8-15 A sales and marketingmanagement magazine conducted a surveyonsalespeople cheatingon their expense reports and other unethicalconduct. In thesurvey on 200 managers, managers have caughtsalespeople cheatingonanexpense report $58 \%$ of the time, workinga second job oncompany time $50 \%$ of the time, listinga "strip bar" as a restauranton anexpense report $22 \%$ of the time, and givinga kickback to a customer $19 \%$ of the time. -As an aid to the establishment of personnel requirements, the director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24-hour period. The director randomly selects 64 different 24-hour periods and determines the number of admissions for each. For this sample, $\overline{X}$ = 19.8 and s² = 25. Estimate the mean number of admissions per 24-hour period with a 95% confidence interval.

TABLE 8-6 A sample of salary offers (in thousands of dollars) given to management majors is: 28, 31, 26, 32, 27, 28, 27, 30, 31, and 29. Using this data to obtain a 95% confidence interval resulted in an interval from 27.5 to 30.3. -Referring to Table 8-6, 95% of the time, the sample mean salary offer to management majors will fall between 27.5 and 30.3.